Subnearrings of $(\mathbb{Z}[x],+,\circ)$

TL;DR

This paper investigates the structure of subnearrings within the nearring of integer polynomials under addition and composition, revealing uncountably many such subnearrings and characterizing those generated by small polynomial sets.

Contribution

It provides an explicit description of subnearrings generated by subsets of \\{1,x,x^2,x^3\\} and demonstrates the uncountability of all subnearrings in this nearring.

Findings

Uncountably many subnearrings exist in the nearring of integer polynomials.

Explicit descriptions are given for subnearrings generated by small polynomial sets.

The structure of subnearrings generated by \\{1,x,x^2,x^3\\} is characterized.

Abstract

We show that the nearring $(\mathbb{Z}[x],+,\circ)$ of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of $\{1,x,x^2,x^3\}$.